# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. Prove that $\overline{\underset{2014\text{ digits}}{\underbrace{111\ldots111}}\underset{2014\text{ digits}}{\underbrace{222\ldots222}}}-\overline{\underset{2014\text{ digits}}{\underbrace{333\ldots333}}}$ is a perfect square. 2. Given a triangle$ABC$with$\widehat{BAC}>90^{0}$and the lengths of its sides are three consecutive even numbers. Find these lengths. 3. Let$a,b$be two positive real numbers such that$a+b$,$ab$are positive integers and$[a^{2}+ab]+[b^{2}+ab]$is a perfect square, where$[x]$is the greatest integer not exceeding$x$. Prove that$a,b$are positive integers. 4. Let$ABC$be an acute triangle with altitudes$AD$,$BE$,$CF$. On the opposite rays of the rays$DA$,$EB$,$FC$choose three points$M,N,P$respectively such that$\widehat{BMC}=\widehat{CNA}=\widehat{APB}=90^{0}$. Prove that the lines containing the sides of the hexagon$APBMCN$are both tangent to a circle. 5. Find all integers$m$such that the equation $x^{3}+(m+1)x^{2}-(2m-1)x-(2m^{2}+m+4)=0$ has an integer solution. 6. Given any triple of real numbers$a,b,c>1$. Prove the following inequality $(\log_{b}a+\log_{c}a-1)(\log_{c}b+\log_{a}b-1)(\log_{a}c+\log_{b}c-1)\leq1.$ 7. Let$ABC$($AB<AC$) be an acute triangle inscribed in a circle$(O)$. The altitudes$AD$,$BE$,$CF$intersect at$H$. Let$K$be the midpoint of$BC$. The tangent lines to the circle$(O)$at$B$and$C$meets at$J$. Prove that$HK$,$JD$,$EF$are concurrent. 8. Find all functions$f:\mathbb{R}\to\mathbb{R}$such that$f$is bounded on a certain interval containing$0$and$f$satisfies $2f(2x)=x+f(x)$ for every$x\in\mathbb{R}$. 9. Let $f(x)=x^{3}-3x^{2}+9x+1964$ be a polynomial. Prove that there exists an integer$a$such that$f(a)$is divisible by$3^{2014}$. 10. Does there exist a continuous funtion$f:\mathbb{R}\to\mathbb{R}$satisfying the following property: for any$x\in\mathbb{R}$, among$f(x)$,$f(x+1)$,$f(x+2)$there are exactly two rational numbers and one irrational number?. 11. Given a sequence$\{a_{n}\}_{1}^{\infty}$where $a_{1}=1,\,a_{2}=2014,\quad a_{n+1}=\frac{2013a_{n}}{n}+\left(1+\frac{2013}{n-1}\right)a_{n-1}.$ Find $\lim_{n\to\infty}\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}\right).$ 12. Let$ABCD$be a quadrilateral circumscribing a circle$(I)$. The sides$AB$and$BC$are tangent to$(I)$at$M$and$N$respectively. Let$E$be the intersection of$AC$and$MN$, and$F$be the intersection of$BC$and$DE$.$DM$intersects$(I)$at another point, say$T$. Prove that$FT$is tangent to$(I)$. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2014 Issue 445
2014 Issue 445
Mathematics & Youth